ll control engineers should be able to detect and iden-tify faults (that is, abnormal conditions in a system) from the analysis of large heterogeneous time-series
data sets. This “Focus on Education” column provides an introduction to multivariable data-based methods for fault detection and fault identification, with the latter being the determination of system variables that contribute the most to a detected fault. For fault identification in statistical process monitoring, the contribution plot is the most com-monly used tool for quickly identifying the most affected variables. Contribution calculations are revisited in the context of principal component analysis (PCA) and T2 sta-tistics, and a two-dimensional (2-D) contribution map is illustrated for the examination of time-series data under faulty conditions. The 2-D contribution map is compared to the traditional one-dimensional (1-D) contribution plot using simulated data from a realistic chemical process. The 2-D contribution map demonstrates the potential to enable a greater understanding of the fault and how its effects are propagated through the system.
INTRODUCTIONFaults inevitably occur in industrial systems and become more prevalent as systems become increasingly large scale and interconnected. The closed-loop performance of the control system depends critically on the proper function-ing of the process and control equipment, so faults need to be detected and diagnosed quickly from the real-time data collected from the system. Rapid detection and diagnosis can minimize downtime, increase the safety of plant oper-ations, and reduce manufacturing costs. Statistical process monitoring (SPM) applies multivariate data-driven meth-ods to process data for fault detection and diagnosis and has been popular in both academic research and indus-trial practice over the past two decades [1]–[5]. Data-driven methods such as PCA, partial least squares (PLS), and other modified methods are used to characterize the data collected during normal process conditions. Such methods are dimensionality reduction techniques that project the high-dimensional process data into much lower dimen-sional spaces. Fault detection is based on multivariate sta-tistics, such as T2 statistics for describing variations within
the lower dimensional space and Q statistics for represent-ing variations in the residual space, in which rigorously derived control limits are computed from the data [1]–[5].
Typical procedures in SPM involve a fault identification step after the detection of a fault to identify the most likely variables closely associated with the fault (that is, the “faulty variables”) by analyzing each variable’s contributions [4]. A contribution plot summarizes quantitative information about the potentially faulty variables. While useful, the tra-ditional contribution plot only examines the contributions at one observation (time point), and multiple contribution plots are needed to illustrate multiple observations in time series data. In comparison, a 2-D contribution map stacks multiple observations into one image to clearly illustrate the contribution of the variables over the entire faulty data times series, which enables the fast identification of faulty variables within large heterogeneous data sets.
The next section is an introduction to PCA [1]–[5], which is the most commonly used technique for fault detection and identification for large heterogeneous data sets. The 2-D contri-bution map is presented as a more effective visualization than the commonly used 1-D contribution plot used for fault iden-tification. The methods are illustrated and compared through application to data collected from a well-known model prob-lem known as the Tennessee Eastman process (TEP).
PCA AND T2 STATISTIC REVISITEDConsider a data matrix X Rm n! # containing m observa-tions of n process variables at the normal process condi-tions. The matrix X should be autoscaled, that is, each pro-cess variable should be pretreated by subtracting its mean and dividing by its standard deviation. PCA dimensional-ity reduction uses the singular value decomposition
,m
X U V1
1 TR-
= (1)
where U Rm m! # and V R nn! # are unitary matrices and Rm n!R # is a diagonal matrix containing the singular val-
ues in decreasing order .0,min m n1 2 g$ $ $ $v v v^ h" ,For each principal component i , its loading vector is given
by the i th column vector of the matrix V , with the vari-ance of the projected training data along the loading vector being equal to .i
2v
In the data-modeling step, only a small number of the principal components, known as the reduction order a , are
Digital Object Identifier 10.1109/MCS.2014.2333295Date of publication: 16 September 2014
» F O C U S O N E D U C AT I O N
Two-Dimensional Contribution Map for Fault Identification
XIAOXIANG ZhU and RIChARD D. BRAATZ
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OctOber 2014 « IEEE CONTROL SYSTEMS MAGAZINE 73
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74 IEEE CONTROL SYSTEMS MAGAZINE » OctOber 2014
retained in the PCA model. Several methods are available for determining the value of a , including the percent vari-ance test, the scree test, and cross validation [1]–[5]. For demonstration purposes, this article uses the percentage variance test, which chooses a based on the lower dimen-sional space containing a specified minimum percentage of the total variance (for example, at least 95%).
Once the reduction order a is determined, the loading matrix P Rn a! # is the first a column vectors in the V matrix.
For an observation x Rn 1! # , the score vector t , which rep-resents the data projection onto the principal components, is
.t P xT= (2)
The T2 statistic, which is a measure of how far the obser-vation is from the center of the characterized normal data, can be calculated directly from the PCA representation by
,T x P P xTa
T2 2R= - (3)
where Raa a!R # is a diagonal matrix containing the first a
rows and columns of R in (1). The threshold for detecting abnormalities in new obser-
vations is given by the T2 statistic
( ) ( )( ) ( )
( , ),T m m aa m m
F a m a1 1
CL2 a =
-
- +-a (4)
where ( , )F a m a-a defines the upper 100a% critical point of the F-distribution with a and m a- degrees of freedom. When the T2 statistics of the new observations (for example, two consecu-tive observations) violate the threshold, a fault is alarmed.
Fault identification is carried out immediately after a fault is detected in the process data using the T2 or any alter-native fault detection statistic. The contribution plot quantifies the contribution of each process variable to the PCA scores (2) to identify the process variables that are most closely associated with, and potentially responsible for or a direct consequence of, the abnormal/out-of-control status. The
procedure for the calculation of the contributions is [4]:1) Given a vector of observations x
(auto-scaled with the mean and variance of the training data) and its calculated score vector t , the contribution of each process vari-able xj to each ti in the score vector t is calculated from
cont( , )
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]]
(5)
where P ,j i is the ,j i^ hth element of the loading matrix .P
2) The total contribution of the pro-cess variable j at the observa-tion is calculated by
CONT( ) cont( , ) .j i ji
a
1==
/ (6)
For each observation, the CONT is a vector whose length is equal to the number of process variables. In
Figure 3 A traditional contribution plot of the testing data set at the time of the detection of Fault #1.
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Observation (h)
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Figure 2 the t2 statistic for normal training data (black) and faultytest data (red) indicates that Fault #1 is detected after hour 25.
OctOber 2014 « IEEE CONTROL SYSTEMS MAGAZINE 75
the traditional contribution plot, the CONT is plotted for the observation at which the fault is detected. For time-series data, the procedure is repeated to generate one con-tribution plot at each observation. A 2-D contribution map, which stacks the series of observations in one single color map, is a more convenient alternative for representing the information. The usefulness of the 2-D contribution map is illustrated in the next section.
TENNESSEE EASTMAN PROCESS EXAMPLEThe TEP is a realistic simula-tion of a chemical facility cre-ated by the Tennessee East-man Company [6]. The TEP is widely used by researchers for evaluating process control and monitoring methods (see, for example, [2], [7], and [8]). Figure 1 shows the TEP flow diagram with a plant-wide control structure, which has 41 measurements, 12 manipu-lated variables, and 21 prepro-grammed faults. The process consists of three main units (a reactor, a separator, and a stripper), and produces two products (labeled G and H) from four reactants (labeled A, C, D, and E). The process is nonlinear, open-loop unsta-ble, and contains a mixture of fast and slow dynamics. The closed-loop system is stable and provided acceptable per-formance over the entire oper-ating regime when no faults occur in the system. Of the 21 preprogrammed faults, some faults are detectable and identifiable using classical single-variable control charts such as Shewhart, exponen-tially weighted moving aver-age (EWMA), and cumulative sum (CUSUM) whereas some faults are challenging for even the most advanced methods. Detailed descriptions of the process and the control struc-ture, as well as a description of the classical control charts, are available in [2] and [9].
In this example, the data under the normal operating conditions were used as the training data for PCA mod-eling, and the data collected during Fault #1 was used as the testing data for demonstrating the method (the data are available online at http://web.mit.edu/braatzgroup/TE_process.zip). Fault #1 is a step change in the A/C feed
Figure 4 Different most-faulty variables were picked out in the traditional contribution plot. (a) One and (b) three samples after the time of the detection of Fault #1.
76 IEEE CONTROL SYSTEMS MAGAZINE » OctOber 2014
ratio in Stream 4 (Figure 1). The data set for normal operat-ing conditions contains 500 observations equally sampled over 25 h, and the data set for Fault #1 has 480 observations equally sampled during a 24-h period. Based on the nor-mal training data, a reduction order a 36= was obtained to retain 95% of the variance. Figure 2, which shows the T2 statistic of the training and testing data sets, indicates a fault detected at around hour 25, several sample times after the fault has occurred. The control limit shown as a blue line was calculated by (4) at the 99% confidence level.
The traditional contribution plot in Figure 3 illustrates the contribution of process variables at the observation upon which the fault was detected (the fault was alarmed after two consecutive T2 control limit violations or eight sampling points after the fault occurrence). The plot sug-gests the fault is most likely associated with process vari-able 16 (XMEAS 16, the stripper pressure). However, con-tribution plots for subsequent observations show different variables having the largest contributions (Figure 4). In this circumstance, because of the dynamics of the closed-loop system, the most critical process variables associated with the fault was indeterminate using the 1-D contribu-tion plots.
The same contribution data plotted as the 2-D contri-bution map in Figure 5 enables the reliable identification
of the key process variables associated with the fault: XMEAS 1 (A feed, Stream 1) and XMV 3 (A feed flow, Stream 1), both of which show consistent strong bands of contribution. Fault #1 is involved with a feed ratio change of A/C in Stream 4, and a control loop changed the A feed in Stream 1 to compensate for the fault. The 2-D contribu-tion map indicates low initial contributions of all process variables at times right after the fault occurs and how the effects of the fault are gradually propagated into XMEAS 1 and XMV 3.
Figure 5 shows that in the first few hours following the fault occurrence (hours 25–29), more than a dozen process variables show high contributions to the fault, which corre-sponds to the period when the closed-loop control system is trying to compensate for the fault. It is unlikely that a control engineer applying 1-D contribution plots to obser-vations in this time period will correctly determine the key faulty variables.
An inspection of Figure 5 indicates that the 1-D contribu-tion plot would correctly identify the variables associated with Fault #1 if the contributions were averaged over 2 h or the data were averaged over two hours before applying the 1-D contribution plot. Averaging over long time windows, however, would directly conflict with the goal of correctly identifying the associated faulty variables quickly after the fault is detected. Further, plotting multiple time series, as in the 2-D contribution map, is generally more useful than plotting single snapshots as done in the 1-D contribution plot because the best time period for identifying faults is not known a priori and will vary depending on the differ-ent fault dynamics.
The 1-D contribution plot will typically give comparable results when the fault response is fast and localized. For ex-ample, consider Fault #4, which is a change in reactor cooling water inlet temperature. The effect of this fault on the vari-ables is simple enough that the contribution is concentrated on XMV 10 (reactor cooling water flow) from the point of fault occurrence, as shown in Figure 6, so the 1-D contribu-tion plot can also quickly identify the faulty variable.
The 2-D contribution map and the 1-D contribution plot are essentially two ways of presenting the same data. In fact, in Figure 5, each column corresponds to the 1-D con-tribution plot at that observation time. The 2-D contribution map, however, assembles the information to enable a more useful visualization for identifying the faulty variables. The 2-D contribution map enables the human operators or control engineers to be better informed, to speed their abil-ity to track down the precise nature and cause of the fault.
The implementation of the 2-D contribution map is briefly summarized by the pseudocode in Table 1.
FINAL REMARKSThe 2-D contribution map provides an alternative means of plotting fault contributions compared to the 1-D contri-bution plot. By plotting multiple observations (time-series
% pretreat the testing data (auto-scaling)
X = pretreat(X );
% calculate the contributions
,m n =6 @ size(X );
cONt = zeros ( ,m n);
for :k m1= % for each new observation k
( , :) ;t P X k ')= l
cont = zeros ,a n^ h for :i a1= % for each principal component
for :j n1= % for each process variable
cont , , ( , ) / ;i j t i X k j P i j i2) ) v=^ ^ ^h h h
cont , cont , cont , ;i j i j i j 0) 2=^ ^ ^^h h h h end
end
, :CONT k =^ h sum(cont, 1);
end
% plot the 2-D contribution map
imagesc(cONt');
x label('New Observations (hour)');
y label('Process Variable');
Table 1 Pseudocode for implementation of the 2-D contribution map.
OctOber 2014 « IEEE CONTROL SYSTEMS MAGAZINE 77
data) on the same map, control engi-neers can more accurately identify the most impacted variables directly and potentially gain a better understand-ing of the fault and how its effects are propagated through the system.
There are alternative means for cal-culating the contributions [5]. While the details of the formulas may be different, the idea of the 2-D contribution map is universally applicable. Complementary to the T2 statistic, the Q statistic, which captures faults in the residual space corresponding to the m a- smallest singular values, can also employ the contribution map for fault identification in a similar manner.
It is the authors’ opinion that every control engineer should receive some training in fault detection and diagno-sis and that the multivariable statistical methods, as well as classical Shewhart, EWMA, and CUSUM control charts, should be the covered at a minimum. Preferably, this content is contained in a course devoted to the topic of fault detection and diagnosis, which should be a component of any undergraduate or graduate control curriculum. If such a course is not offered, then the con-tent should be covered in one to two lectures of an introductory controls, systems engineering, or data analysis course. At the authors’ institution, this material is covered in the modeling and data analysis section of an introduc-tory systems engineering course mostly taken by first-year graduate students.
ACKNOWLEDGMENTSThe authors gratefully acknowledge BP for funding.
REFERENCES[1] S. J. Qin, “Survey on data-driven industrial process monitoring and diagnosis,” Annu. Rev. Contr., vol. 36, no. 2, pp. 220–234, 2012.[2] L. H. Chiang, E. L. Russell, and R. D. Braatz, Data-Driven Methods for Fault Detection and Diagnosis in Industrial Systems. London: Springer-Verlag, 2000.[3] S. J. Qin, “Statistical process monitoring: Basics and beyond,” J. Che-momet., vol. 17, nos. 8–9, pp. 480–502, 2003.[4] J. A. Westerhuis, S. P. Gurden, and A. K. Smilde, “Generalized contribu-tion plots in multivariate statistical process monitoring,” Chemomet. Intell. Lab. Syst., vol. 51, no. 1, pp. 95–114, 2000.
[5] T. Kourti and J. F. MacGregor, “Multivariate SPC methods for process and product monitoring,” J. Qual. Technol., vol. 28, no. 4, pp. 409–428, 1996.[6] J. J. Downs and E. F. Vogel, “A plant-wide industrial process control problem,” Comput. Chem. Eng., vol. 17, no. 3, pp. 245–255, 1993.[7] N. L. Ricker and J. H. Lee, “Nonlinear model predictive control of the Tennessee Eastman challenge process,” Comput. Chem. Eng., vol. 19, no. 9, pp. 961–981, 1995.[8] T. J. McAvoy and N. Ye, “Base control for the Tennessee Eastman prob-lem,” Comput. Chem. Eng., vol. 18, no. 5, pp. 383–413, 1994.[9] P. R. Lyman and C. Georgakis, “Plant-wide control of the Tennessee Eastman problem,” Comput. Chem. Eng., vol. 19, no. 3, pp. 321–331, 1995.
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Figure 5 A 2-D contribution map of the testing data (Fault #1) provides a visual identifica-tion of the most contributing variables in bright bands.
